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Fbgbkl_jkl\hh[s_]hbijhn_kkbhgZevgh]h h[jZah\ZgbyJhkkbckdhcN_^_jZpbb
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2
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3
Hij_^_e_gb_ Nmgdpby fgZau\Z_lkyg_ij_ju\ghc\lhqd_ x_keb^eyex[h]hε > kms_kl\m_lδ=δ(ε) >lZdh_qlhbalh ]hqlh| x - x’| < δ , ke_^m_lqlh| f(x) - f(x’)| < ε. <\_^_ggh_\ur_hij_^_e_gb_^himkdZ_lfh^bnbdZpbxm^h[ gmx^ey^Zevg_cr_]hbaeh`_gby Hij_^_e_gb_’. NmgdpbyfgZau\Z_lkyg_ij_ju\ghc\lhqd_ x_keb^eyex[hchdj_klghklbUlhqdbf(x)kms_kl\m_lhdj_kl ghklvVlhqdbxlZdZyqlhbalh]hqlhlhqdZx’ijbgZ^e_`bl Vke_^m_lqlhf(x’) ijbgZ^e_`blU. G_ljm^gh\b^_lvqlh^eyqbkeh\uonmgdpbcHij_^_e_gb_b Hij_^_e_gb_’wd\b\Ze_glguihkdhevdmkh^ghcklhjhgufgh`_ kl\hlhq_d{x’} lZdboqlh| x - x’| < δy\ey_lkyhdj_klghklvxlhqdb xgZau\Z_fhcδ-hdj_klghklvxx khhl\_lkl\_gghfgh`_kl\hlhq_d {y} lZdboqlh| f(x) - y| < εy\ey_lkyhdj_klghklvxlhqdbf(x), gZ au\Z_fhcε-hdj_klghklvxf(x) Zk^jm]hcklhjhgu\gmljbex[hc hdj_klghklb U lhqdb f(x) kh^_j`blky ε-hdj_klghklv ^ey ^hklZ lhqgh fZeh]h ε khhl\_lkl\_ggh \ ex[hc hdj_klghklb V lhqdb x kh^_j`blkyδ-hdj_klghklv^ey^hklZlhqghfZeh]hδ). JZkkfhljbf^\Zfgh`_kl\ZXb Y=h\hjylqlhaZ^Zghhlh [jZ`_gb_F: X → Y_kebaZ^ZghijZ\beh aZdhg ihdhlhjhfmdZ` ^hfmwe_f_glmxba XihklZ\e_g\khhl\_lkl\b_we_f_gly = F(x) ba Y Qbkeh\Zy nmgdpby y\ey_lky gZb[he__ ba\_klguf ijbf_jhf hlh[jZ`_gby<wlhfkemqZ_h[uqghX = Y = R -fgh`_kl\h\_s_kl \_gguoqbk_e qbkeh\ZyijyfZy ZaZdhgFaZ^Z_lkynhjfmehc:gZ ijbf_j \_s_kl\_gghfm qbkem x klZ\blky \ khhl\_lkl\b_ \_s_kl \_ggh_qbkeh sin x \wlhfkemqZ_F _klvnmgdpbykbgmk
4
Ihgylb_hlh[jZ`_gbyhij_^_e_gh^eyex[hciZjuijhba\hev guofgh`_kl\H^gZdhfh`gheb\ijhba\hevghfkemqZ_^Zlvhij_ ^_e_gb_g_ij_ju\ghklbFihZgZeh]bbkHij_^_e_gb_fbebHi j_^_e_gb_f’?G_ljm^gh\b^_lvqlhwlh]hk^_eZlvg_evayihkdhev dm gZ ijhba\hevguo fgh`_kl\Zo g_l gb ihgylby hdj_klghklb bk ihevam_fh]h \ Hij_^_e_gbb ’ gb ihgylby δ-hdj_klghklb εhdj_klghklb bkihevam_fh]h\Hij_^_e_gbbLZdqlh^ey\\_^_gby dhjj_dlgh]h hij_^_e_gby ihgylby g_ij_ju\ghklb F fu ^he`gu eb[h\\_klbij_^\Zjbl_evghihgylb_hdj_klghklb\hh[s_eb[hih gylb_ ε-hdj_klghklb GZ ijbf_j_ qbkeh\uo nmgdpbc \b^gh qlh εhdj_klghklby\eyxlkyqZklgufkemqZ_fhdj_klghkl_c\hh[s_b_k ebfuohlbf^ZlvgZb[he__h[s__hij_^_e_gb_g_ij_ju\ghklbfu ^he`gu khkj_^hlhqblv k\h_ \gbfZgb_ gZ dhjj_dlghf \\_^_gbb ih gylbyijhklhhdj_klghklblhqdb\ijhba\hevghffgh`_kl\_ Fgh`_kl\hgZdhlhjhf“ijZ\bevgh” \\_^_ghihgylb_hdj_kl ghklb gZau\Z_lky lhiheh]bq_kdbf ijhkljZgkl\hf Ih^q_jdg_f lj_[h\Zgb_qlh[ufgh`_kl\h[uehlhiheh]bq_kdbfijhkljZgkl\hf y\ey_lkyfbgbfZevguf^eylh]hqlh[u[uehdhjj_dlghhij_^_e_gh ihgylb_g_ij_ju\gh]hhlh[jZ`_gby 1. HIJ?>?E?GB?LHIHEH=BQ?KDH=HIJHKLJ:GKL<:B I?J
5
bgl_j\Zekp_gljhf\wlhclhqd_ l_ε-hdj_klghklv^ey^hklZlhqgh fZeh]h ε p_ebdhf \oh^bl \ wlh fgh`_kl\h >ey hldjuluo fgh `_kl\\uihegyxlky^\Z\Z`guok\hckl\Z h[t_^bg_gb_ex[h]h ^Z `_[_kdhg_qgh]h gZ[hjZhldjuluofgh`_kl\_klvhldjulh_fgh`_ kl\h b i_j_k_q_gb_ dhg_qgh]h qbkeZ hldjuluo fgh`_kl\ _klv hl djulh_fgh`_kl\h HdZau\Z_lky_kebg_dhlhjucgZ[hjfgh`_kl\h[eZ^Z_lwlbfb k\hckl\Zfblhkfgh`_kl\ZfbbamdZaZggh]hgZ[hjZfh`ghjZ[hlZlv \hfgh]hflZd`_dZdkh[uqgufbhldjulufbfgh`_kl\Zfb>Z^bf lhqgu_hij_^_e_gby JZkkfhljbfijhba\hevgh_fgh`_kl\h; Hij_^_e_gb_GZ[hj τih^fgh`_kl\fgh`_kl\ZXgZau \Z_lkylhiheh]b_c_kebhgh[eZ^Z_lke_^mxsbfbk\hckl\Zfb L ;bimklh_fgh`_kl\h\oh^yl\ τ LL h[t_^bg_gb_ex[h]hqbkeZ fgh`_kl\ ba τ ijbgZ^e_`bl τ LLL i_j_k_q_gb_ dhg_qgh]h qbkeZ fgh`_kl\baτijbgZ^e_`blτ?kebgZ[hjτaZ^Zg;gZau\Z_lky lhiheh]bq_kdbfijhkljZgkl\hfZ\oh^ysb_\τfgh`_kl\ZgZau \Zxlkyhldjulufb Ijbf_jhf lhiheh]bq_kdh]h ijhkljZgkl\Z y\ey_lky qbkeh\Zy ijyfZy k fgh`_kl\Zfb hldjulufb \ h[uqghf kfuke_ >_ckl\b l_evgh \ky qbkeh\Zy ijyfZy hq_\b^guf h[jZahf hldjulZ imklh_ fgh`_kl\h\dexqZxl\qbkehhldjuluoihhij_^_e_gbx wlhg_ijh lb\hj_qb\h - ihkdhevdm \ imklhf fgh`_kl\_ g_l lhq_d lh fh`gh kqblZlvqlhdZ`^Zybagbo \oh^bl\imklh_fgh`_kl\hkg_dhlh jhc ε-hdj_klghklvx DZd m`_ kdZaZgh \ur_ LL b LLL \uiheg_gu Lhiheh]bx khklhysmx ba h[uqguo hldjuluo fgh`_kl\ gZ qbkeh \hcijyfhc[m^_fgZau\Zlvh[uqghclhiheh]b_c
6
Ijb\_^_f_s_^\Zijbf_jZGZex[hf;jZkkfhljbflhiheh ]bx\dhlhjhc\k_]h^\Zfgh`_kl\Z \k_;bimklh_LZdZylhiheh ]bygZau\Z_lkyljb\bZevghcIjhlb\hiheh`gZykblmZpby-gZex[hf ;\dexqbf\lhiheh]bx\hh[s_\k_ih^fgh`_kl\Z; \qZklghklb \k__]hlhqdb kZfh;bimklh_ih^fgh`_kl\hWlZlhiheh]bygZ au\Z_lky^bkdj_lghc H[jZlbl_ \gbfZgb_ qlh ljb\bZevgmx b ^bkdj_lgmx lhiheh ]bx fu aZ^Zeb hibkZ\ \k_ \oh^ysb_ \ gbo fgh`_kl\Z K h[uqghc lhiheh]b_cfug_kfh]ebwlhk^_eZlvbgZfijbrehkvhibku\Zlv__ kihfhsvxk\hckl\Zdhlhjhfmm^h\e_l\hjyxl__fgh`_kl\ZQlh[u ba[_`Zlvwlh]hg_m^h[kl\Z[ueh\\_^_ghihgylb_[Zaulhiheh]bb Hij_^_e_gb_GZ[hjhldjuluofgh`_kl\SgZau\Z_lky [Zahc lhiheh]bb τ _keb ex[h_ fgh`_kl\h ba τ _klv \hafh`gh [_kdhg_qgh_ h[t_^bg_gb_fgh`_kl\baS. ;Zahch[uqghclhiheh]bbgZijyfhcy\eyxlkyε-hdj_klghklb >_ckl\bl_evgh h[uqgh_ hldjulh_ fgh`_kl\h oZjZdl_jbam_lky l_f qlh dZ`^Zy _]h lhqdZ bf__l g_dhlhjmx ε-hdj_klghklv \oh^ysmx \ wlhfgh`_kl\hLZdqlhhq_\b^ghqlhkZfhfgh`_kl\h_klvh[t_^b g_gb_mdZaZgguoε-hdj_klghkl_c\k_o_]hlhq_d Ijb\_^_f_s_^\Zijbf_jZI_j\ucbagbo-lhiheh]byAZjbk kdh]h gZ qbkeh\hc ijyfhc - bgl_j_k_g djhf_ \k_]h ijhq_]h l_f qlh\hagbd\j_ZevghcfZl_fZlbq_kdhcaZ^Zq_Zg_dZdwdahlbq_kdbc ijbf_j^eymq_[gbdZ<wlmlhiheh]bx\dexq_gu\kyijyfZybimk lh_ fgh`_kl\h Z lZd`_ \k_ fgh`_kl\Z gZ ijyfhc ^hiheg_gby ^h dhlhjuokhklhylbadhg_qgh]hqbkeZlhq_d Ke_^mxsZy lhiheh]by gZ qbkeh\hc ijyfhc khklhbl ba \k_c ijyfhcbimklh]hfgh`_kl\ZZlZd`_\k_ohldjuluobgl_j\Zeh\\b ^Z a, +∞ ]^_a -lhqdZijyfhc WlZlhiheh]bygZau\Z_lkyijZ\hc
7
Hlf_lbfqlh\lhqghklbZgZeh]bqghfh`ghaZ^Zlvbe_\mxlhiheh ]bx Lhiheh]byfh`_lgZke_^h\ZlvkyGZijbf_j\iehkdhklbbf_ _lkylhiheh]bykhklhysZybah[uqguohldjuluofgh`_kl\ ZgZeh ]bqgh kemqZx qbkeh\hc ijyfhc Lh]^Z gZ e_`Zs_c \ iehkdhklb ijyfhc \hagbdZ_l lhiheh]by \ dhlhjhc hldjulufb fgh`_kl\Zfb y\eyxlkyi_j_k_q_gbykwlhcijyfhcfgh`_kl\hldjuluo\iehkdh klb WlZ lhiheh]by gZau\Z_lky bg^mpbjh\Zgghc < jZkkfZljb\Z_ fhfijbf_j_bg^mpbjh\ZggZylhiheh]by-wlhh[uqgZylhiheh]bygZ ijyfhc <g_dhlhjuokemqZyojZaebqgu_lhiheh]bbgZh^ghfblhf`_ fgh`_kl\_fh`ghkjZ\gb\Zlvf_`^mkh[hc=h\hjylqlhlhiheh]by τgZXkbevg__lhiheh]bbσgZlhf`_fgh`_kl\__keb\k_fgh `_kl\Z\oh^ysb_\σ\oh^yllZd`_b\τHq_\b^ghqlhex[Zylh iheh]by kbevg__ q_f ljb\bZevgZy Z ^bkdj_lgZy - kbevg__ ex[hc lhiheh]bbLZd`_ihgylghqlhh[uqgZylhiheh]bykbevg__q_flh iheh]byAZjbkkdh]hbq_fijZ\Zylhiheh]byb\lh`_\j_fylhiheh ]bxAZjbkkdh]hbijZ\mxlhiheh]bxkjZ\gblvf_`^mkh[hcg_evaygb h^gZ ba gbo g_ y\ey_lky [he__ kbevghc q_f ^jm]Zy [he__ lh]h ^hdZ`bl_ qlh _keb g_dhlhjh_ fgh`_kl\h qbkeh\hc ijyfhc \oh^bl kjZam \ h[_ wlb lhiheh]bb lh wlh eb[h \ky qbkeh\Zy ijyfZy eb[h imklh_fgh`_kl\h Hij_^_e_gb_ Hdj_klghklvxlhqdb\lhiheh]bq_kdhfijh kljZgkl\_ gZau\Z_lky ex[h_ hldjulh_ fgh`_kl\h kh^_j`Zs__ mdZaZggmxlhqdm Hq_\b^gh qlh \ h[uqghc lhiheh]bb ihgylb_ hdj_klghklb m^h\e_l\hjy_l^Zgghfmhij_^_e_gbx
8
Bkihevamy \\_^_ggh_ hij_^_e_gb_ hdj_klghklb g_ljm^gh ^h dZaZlv ke_^mxs__ k\hckl\h hldjuluo fgh`_kl\ ex[h]h lhiheh]b q_kdh]hijhkljZgkl\Z fgh`_kl\hAhldjulhlh]^Zblhevdhlh]^Z dh]^ZdZ`^ZylhqdZxbaAbf__lhdj_klghklvp_ebdhf\oh^ysmx \ A >hdZ`bl_ wlh ml\_j`^_gb_ kZfhklhyl_evgh H[jZlbl_ \gbfZ gb_qlhoZjZdl_jbklbq_kdh_k\hckl\hh[uqguohldjuluofgh`_kl\ gZqbkeh\hcijyfhcy\ey_lkyqZklgufkemqZ_fwlh]hml\_j`^_gby G?IJ?JU:QBLHIHEH=BB ImklvaZ^Zghhlh[jZ`_gb_F: X → Y]^_XbY -lhiheh]bq_ kdb_ijhkljZgkl\Zklhiheh]byfbkhhl\_lkl\_gghτbσIhkdhev dmfu\\_ebhij_^_e_gb_hdj_klghklblhqdb\lhiheh]bq_kdhfijh kljZgkl\_ fu fh`_f ^Zlv hij_^_e_gb_ g_ij_ju\ghklb F \ lhqd_ iheghklvxZgZeh]bqghHij_^_e_gbx’. Hij_^_e_gb_Hlh[jZ`_gb_FgZau\Z_lkyg_ij_ju\guf\ lhqd_x∈X_keb^eyex[hchdj_klghklbU∈σlhqdbf(x)\Ykm s_kl\m_l hdj_klghklv V∈τ lhqdb x \ X lZdZy qlh ba lh]h qlhlhqdZx’ijbgZ^e_`blVke_^m_lqlhf(x’) ijbgZ^e_`blU. Hij_^_e_gb_ 6.Hlh[jZ`_gb_g_ij_ju\gh_\dZ`^hclhqd_ x fgh`_kl\Z XgZau\Z_lkyg_ij_ju\gufgZX. < kemqZ_ dh]^Z fgh`_kl\h X aZnbdkbjh\Zgh fu [m^_f gZ au\Zlvhlh[jZ`_gbyijhklhg_ij_ju\gufbg_mdZau\ZyX. G_ij_ju\gu_ hlh[jZ`_gby oZjZdl_jbamxlky ke_^mxsbf k\hckl\hf L_hj_fZ 1. Hlh[jZ`_gb_F: X → Yg_ij_ju\ghlh]^Zblhev dh lh]^Z dh]^Z ^ey ex[h]h hldjulh]h fgh`_kl\Z U∈σ ijh kljZgkl\ZY_]hijhh[jZaV=F-1(U)ijbgZ^e_`bl τl_y\ey _lkyhldjuluffgh`_kl\hflhiheh]bq_kdh]hijhkljZgkl\ZX.
9
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10
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11
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12
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13
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14
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15
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16
EBL?J:LMJ: 1. <\_^_gb_\lhiheh]bx/;hjbkh\bqX=;ebagydh\GFBajZb e_\bqY:b^jFhkd\Z GZmdZNbafZlebl -k 2. Klbgjh^ G Qbgg M I_j\u_ ihgylby lhiheh]bb Fhkd\Z Fbj 1967.-k 3. NjZgkbk >` Dgb`dZ k dZjlbgdZfb ih lhiheh]bb Fhkd\Z Fbj 1991.-k 4. Nmdk>;Nhf_gdh:L=ml_gfZo_j<E=hfhlhibq_kdZylhih eh]byFhkd\Z Ba^Zl_evkl\hF=M -k
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